Nuclear Instruments and Methods in Physics Research A 733 (2014) 80–85

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Nuclear Instruments and Methods inPhysics Research A

0168-90http://d

☆Resen CorrE-m

journal homepage: www.elsevier.com/locate/nima

Effects of beam-plasma instabilities on neutralized propagationof intense ion beams in background plasma$

Edward A. Startsev n, Igor D. Kaganovich, Ronald C. DavidsonPlasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA

a r t i c l e i n f o

Available online 3 June 2013

Keywords:FusionPlasmaHeavy-ion beamTwo-stream instabilityFocusing

02/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.nima.2013.05.090

arch supported by the U.S. Department of Enesponding author.ail address: estartsev@pppl.gov (E.A. Startsev)

a b s t r a c t

The streaming of an intense ion beam relative to background plasma can cause the development of fastelectrostatic collective instabilities. The plasma waves produced by the two-stream instability modify theion beam current neutralization and produce non-linear average forces which can lead to defocusing ofthe ion beam. Recently, a theoretical model describing the average de-focusing forces acting on the beamions has been developed, and the scalings of the forces with beam-plasma parameters have beenidentified (Startsev et al. in press[1]). These scalings can be used in the development of realistic ion beamcompression scenarios in present and next-generation ion-beam-driven high energy density physics andheavy ion fusion experiments. In this paper the results of particle-in-cell simulations of ion beampropagation through neutralizing background plasma for NDCX-II parameters are presented. Thesimulation results show that the two-stream instability can play a significant role in the ion beamdynamics. The effects of velocity tilt on the development of the instability and ion beam compressibilityfor typical NDCX-II parameters are also simulated. It is shown that the two-stream instability may be animportant factor in limiting the maximum longitudinal compression of the ion beam.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction and theoretical model

In ion-beam-driven high energy density physics and heavy ionfusion applications, the intense ion beam pulse propagatesthrough a background plasma before it is focused onto the target[1,2]. The streaming of the ion beam relative to the backgroundplasma can cause the development of fast electrostatic collectiveinstabilities [3]. These instabilities produces fluctuating electro-static fields that cause a significant drag on the background plasmaelectrons and can accelerate electrons up to the average ion beamvelocity. Consequently, the dominant electron current can reversethe beam self-magnetic field. As a result, the magnetic self-fieldforce reverses sign and contributes to a transverse defocusing ofthe beam ions, instead of a pinching effect in the absence ofinstability [4,5]. In addition, the ponderomotive force of theunstable waves push background electrons transversely away fromthe unstable region inside the beam, which creates an ambipolarelectric field, which also leads to ion beam transverse defocusing.Because the instability is resonant it is strongly affected and thuscan be effectively mitigated and controlled by the longitudinalfocusing of the ion beam [6–8].

ll rights reserved.

ergy.

.

Two-stream collective interactions between the beam ions andplasma electrons excite unstable waves with phase velocity ω=kzslightly below the ion beam velocity vb. Here ω is the unstablewave frequency and kz is the unstable wave longitudinal wave-number (along the beam propagation direction z). In a recentpublication [1] we have studied the nonlinear effects of a devel-oped two-stream instability on a non-relativistic heavy ion beamwith ions of mass mb, charge q¼ Zbe, density nb, and longitudinalvelocity vb≪c during its propagation through cold neutralizingbackground plasma with density np and with no externally appliedmagnetic field. Here c is the speed of light. There, we developed amodel that qualitatively captures the main nonlinear features thathave been observed in numerical simulations of a proton beampropagating through a dense plasma background, and providesorder-of-magnitude estimates of the average forces acting on thebeam ions, provided that Zbnb=np≪1. Here we give a morecomplete elaboration of this model.

Due to the large beam ion massmb≫me, whereme is the electronmass, and for beams that are much longer than the resonantwavelength of the beam-plasma two-stream instability lb≫vb=ωpe,where lb is the beam length and ωpe ¼ ð4πe2np=meÞ1=2 is the electronplasma frequency, the result of the instability is to produce short-wavelength electrostatic wave perturbations with frequency close tothe plasma frequency ω≈ωpe. The unstable waves have a phasevelocity which is close to the beam velocity, ω=kz≈ωp=kz≈vb. In whatfollows we assume that the beam transverse dimensions have a

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E.A. Startsev et al. / Nuclear Instruments and Methods in Physics Research A 733 (2014) 80–85 81

single characteristic length of order of the beam radius rb which isnot much smaller than electron collisionless skin-depth, rb≳c=ωpe. Inthis case kzrb≫1 and the instability is purely longitudinal. The radialdependence appears in all equations as a variable.

The unstable wave growth saturates either by longitudinal trap-ping of plasma electrons in the wave, which happens when theelectron's amplitude of velocity oscillation in the wave become oforder of the phase velocity of the wave vem∼ω=kz∼vb, or by trapping ofbeam ions, which happens when the beam ion's amplitude ofvelocity oscillation in the wave becomes of order of the differencebetween the beam velocity and the phase velocity of the wavevbm∼vb−ω=kz∼γ=kz≈ðγ=ωpeÞvb. Here we used the fact that ω−kzvb∼γ,where γ ¼ Im ω∼ωpeðωpb=ωpeÞ2=3 is the linear growth rate of theinstability, and ωpb ¼ ð4πZ2be2nb=mbÞ1=2 is the plasma frequency ofthe beam ions [9]. Because the electrons and beam ions bothexperience the same wave electric field, their oscillation velocitiesare related by meωvem ¼mbðω−kzvbÞvbm=Zb, or equivalently, vem∼ðmb=ZbmeÞðγ=ωpeÞvbm. Therefore, if the beam ions become trapped by thewave before the electrons, vbm∼ðγ=ωpeÞvb and the correspondingamplitude of the electron velocity oscillations has to satisfy therelation vem∼ðmb=ZbmeÞðγ=ωpeÞ2vb≲vb, or ½Zbðmb=meÞðnb=npÞ2�1=3≲1.Otherwise, if ½Zbðmb=meÞðnb=npÞ2�1=3≳1, the electron velocity oscilla-tion amplitude becomes vem∼vb first, and the saturation is determinedby electron trapping. If the saturation is caused by beam ion trapping,the beam density becomes highly modulated in the longitudinaldirection. The beam splits longitudinally into short bunches withlength ∼vb=ωpe≪lb.

The electric field of the waves also generates average transverseforces on the beam ions. An average ambipolar electric field is setup which acts against the Lorentz force and ponderomotivepressure of the wave exerted on the cold plasma electrons

Fig. 1. Plots of ion beam density; beam density profile along nbðx¼ 0; z; tÞ plotted versust ¼ 200 ns (bottom).

according to

e〈Ex〉∼e〈vez〉c

〈By〉þme ðvemÞ24rb

; ð1Þ

where rb is the beam radius. At the same time, the wave electric fieldproduces an average nonlinear electron longitudinal current density〈jnonz 〉¼ −e〈δneδvez〉≈−enp〈ðδvezÞ2〉=vb≈−enpðvemÞ2=2vb in addition to thebeam current density jbz . Here we used the linear relationδne≈npðkz=ωÞδvez≈npδvez=vb. The total injected current density jbz þ〈jnonz 〉 will produce an inductive plasma response current density〈jindz 〉¼ −enp〈vez〉, which will reduce the total injected current densityby the factor 1=ð1þ r2b=λ2peÞ, where λpe ¼ c=ωpe is the collisionless skin-depth. Therefore the total current density in the plasma becomes

〈jz〉≈jbz

ð1þ r2b=λ2peÞ1−

12

npZbnb

�vemvb

�2" #; ð2Þ

and the associated azimuthal self-magnetic field is

〈By〉∼2πc〈jz〉rb∼

2πZbenbrbβbð1þ r2b=λ2peÞ

1−12

npZbnb

vemvb

�2 #:

"ð3Þ

Note fromEqs. (2) and (3) that thepresence ofwaves in theplasmadueto the instability can cause the reversal of the total longitudinal currentdensity and associated azimuthal self-magnetic field ifðvem=vbÞ24 ð2Zbnb=npÞ. The average electron flow produced by theinductive electric field (〈vez〉-0 as c-∞) is given by

〈vez〉≃Zbnbnp

vbð1þ λ2pe=r2bÞ

1−12

npZbnb

�vemvb

�2" #: ð4Þ

z at t ¼ 50 ns and t ¼ 200 ns (top); beam density (x,z) contour plots at t ¼ 50 ns and

E.A. Startsev et al. / Nuclear Instruments and Methods in Physics Research A 733 (2014) 80–8582

The total average transverse force acting on the beam ions isgiven by

Fx ¼ q〈Ex〉−q vbc 〈By〉

≃Zbeð〈vez〉−vbÞ

c〈By〉þmeZb

ðvemÞ24rb

≃meZbv2b2rb

!Zbnbnp

� �þ xþ xð1þ λ2pe=r2bÞ

þ x2

ð1þ λ2pe=r2bÞ2

#;

"ð5Þ

where x¼ ð1=2Þðvem=vbÞ2−Zbnb=np.The saturated level of the two-stream instability determines

the magnitude of the average transverse force when ð1=2Þðvem=vbÞ2≫Zbnb=np, or equivalently, the beam density is sufficientlylarge ðZbnb=npÞ2=3≫ðZbme=mbÞ4=3. In that case x40, and the forceis defocusing. For beams with rb≫λpe, the force on the beam ions isgiven by

Fx≃meZbv2b2rb

!½ðxþ 1Þ2−1�; ð6Þ

where x¼ ð1=2Þðvem=vbÞ2. For ðZbnb=npÞ2=3≳ðZbme=mbÞ1=3, thesaturation is due to electron trapping, and x∼1 andFx∼meZbv2b=rb, while for low-density beams with ðZbme=mbÞ4=3≪ðZbnb=npÞ2=3≲ðZbme=mbÞ1=3, the saturation is by beam ion trappingand x∼ðnb=npÞ4=3ðZbmb=meÞ2=3≲1, and the force is correspondinglysmaller with Fx∼meZbv2bðnb=npÞ4=3ðZbmb =meÞ2=3=rb. Given thetransverse defocusing forces on the beam ions in Eq. (6), we canestimate the defocusing time T∼ðmbrb=FxÞ1=2 it takes the beamradius to double after the instability has developed and saturated,i.e., Δrb=rb≃1 and the defocusing propagation distance is L¼ vbT .

Fig. 2. Plots of beam ions (left) and plasma electrons (ri

Fig. 3. Plots of total current density jz at r¼0 and transv

For sufficiently large beam density that ðZbnb=npÞ2=3≳ðZbme=mbÞ1=3,the defocusing time and distance are

T∼rbvb

� ��mbZbme

�1=2; L∼rb

�mbZbme

�1=2; ð7Þ

while for low density beams with ðZbme=mbÞ4=3≪ðZbnb=npÞ2=3≲ðZbme=mbÞ1=3, the defocusing time and distance are

T∼rbvb

� ��npZbnb

�2=3�mbZbme

�1=6;

L∼rb

�npZbnb

�2=3�mbZbme

�1=6: ð8Þ

Note that the total defocusing time should also include the time forthe instability to grow and saturate, Tsat ¼ ð1=γÞln L, where L dependson details of the beam density profile, and is a large number forsufficiently long, smooth beam profiles with ωpelb=vb≫1.

2. Numerical simulations of NDCX-II experiment

In this section we present the results of numerical simulationsusing the particle-in-cell code LSP [10] of an intense ion beampropagating through neutralizing plasma with parameters rele-vant to the NDCX-II experiment [2]. The simulations were carriedout in slab geometry ∂=∂y¼ 0 with electric and magnetic fieldpolarizations given by E¼ Exêx þ Ezêz and B¼ Byêy. The simula-tions were carried out for the following parameters: a lithium ionbeam with Gaussian density profile in both the longitudinal and

ght) longitudinal phase-space z−vzγ=c at t ¼ 200 ns.

erse electric field Ex at z¼ 41 cm at time t ¼ 200 ns.

Fig. 4. Contour plots in (x,z) of self-magnetic field By at t ¼ 50 ns and t ¼ 200 ns.

Fig. 5. Plots of ion beam density: beam density profile nbðx¼ 0; z; tÞ plotted versus z at t ¼ 100 ns, t ¼ 200 ns and t ¼ 216 ns (left); beam density (x,z) contour plots att ¼ 100 ns, t ¼ 200 ns and t ¼ 216 ns (right).

E.A. Startsev et al. / Nuclear Instruments and Methods in Physics Research A 733 (2014) 80–85 83

E.A. Startsev et al. / Nuclear Instruments and Methods in Physics Research A 733 (2014) 80–8584

transverse directions and pulse duration T ¼ 12 ns. The beamvelocity is vb ¼ c=30; the beam density is nb ¼ 2� 109 cm−3; andthe beam radius is rb ¼ 1:41 cm. The beam propagates through astationary, singly-ionized carbon plasma with plasma densitynp ¼ 0:55� 1011 cm−3. For these parameters, rb∼λpe ¼ c=ωpe andthe characteristic linear exponentiation time of the two-streaminstability is Γ−1 ¼ ðIm ωÞ−1 ¼ 4:1 ns.

The results of the simulations are illustrated in Figs. 1–6. Attime t¼0 the ion beam enters the plasma at z¼0. The changes inion beam density profile during the beam propagation are shownin Fig. 1. At time t ¼ 50 ns the two-stream instability between thebeam ions and the plasma electrons begins to develop [Fig. 1(left)]and has completely developed to the saturation level by timet ¼ 200 ns [Fig. 1(right)] (or two meters of propagation in theplasma). By this time, the beam density is modulated longitudin-ally with density variations of order 100% of the original beamdensity. This occurs because for the simulation parameters½Zbðmb=meÞðnb=npÞ2�1=3≈2:6 and therefore the trapping of both thebeam ions and background electrons is responsible for saturationas shown in Fig. 2, where the longitudinal phase space z−vzγ=c isplotted for the beam ions and plasma electrons at t ¼ 200 ns. Plotsof the average total current density o jz4 are shown in Fig. 3 attime t ¼ 200 ns at the beam center x¼0, and plots of thetransverse profile of average transverse electric field oEx4 att ¼ 200 ns at the location of the maximum current density(z¼ 41 cm), which also correspond to the maximum amplitudeof plasma oscillations.

The azimuthal magnetic field By is plotted in Fig. 4 at t ¼ 50 ns(before saturation) and t ¼ 200 ns (after saturation). Note that as the

Fig. 6. Plots of beam ion longitudinal phase-space z−v

instability develops and saturates, the magnetic field changes signbecause the current density ojz4 becomes negative. Both theamplitude and the sign of the total current density and theamplitude and the sign of the ambipolar average transverse electricfield are consistent with the theoretical estimates in Section 1. In theNDCX-II compression experiments, the beam enters the plasma witha velocity tilt Δvb=vb ¼ 0:1, which should produce longitudinal beamcompression after t ¼ 220 ns. Note that even with zero velocity tilt,the two-stream instability leads to beam break-up at t ¼ 200 ns, butno transverse defocusing Fig. 1. With velocity tilt Δvb=vb ¼ 0:1, theinstability does not develop enough to break-up the beam andprevent the longitudinal focusing.

Fig. 5 shows plots of the beam density profiles at differenttimes during compression. A maximum longitudinal compressionfactor of C¼67 is achieved at t ¼ 216 ns. Compression is limited bythe instability spoiling the beam longitudinal phase space asshown in Fig. 6. The beam length at maximum compressioncorresponds to the size of the beam region where the initial linearvelocity tilt was significantly spoiled by the instability, which (forthis simulation) corresponds to one wavelength of the two-streaminstability ∼vb=ωpe [Fig. 6]. Hence, the maximum compression waslimited to C∼Tωpe∼100.

3. Conclusions

The streaming of an intense ion beam relative to backgroundplasma can cause the development of fast electrostatic collectiveinstabilities. The present simulations of an intense ion beam

zγ=c at t ¼ 100 ns, t ¼ 200 ns and t ¼ 216 ns.

E.A. Startsev et al. / Nuclear Instruments and Methods in Physics Research A 733 (2014) 80–85 85

propagating though neutralizing plasma with parameters relevantto the NDCX-II experiment show that the instabilities producefluctuating electrostatic fields that cause a significant drag on thebackground plasma electrons and can accelerate electrons up tovelocities comparable to the ion beam velocity. Consequently, the(strong) electron current reverses the direction of the beam-induced self-magnetic field. As a result, the magnetic self-field forcereverses sign and leads to a transverse defocusing of the beam ions,instead of a pinching effect in the absence of instability. In addition,the ponderomotive force [Eq. (1)] of the unstable waves pushes thebackground electrons transversely away from the unstable regioninside the beam, which creates an ambipolar electric field, whichalso leads to a transverse defocusing of the beam ions [Eq. (5)]. Thesimple theoretical model that is presented in Section 1 providesorder-of-magnitude estimates of the nonlinear defocusing forcesthat develop at saturation of the two-stream instability. In oursimulations for beam parameters typical of NDCX-II, the nonlineardefocusing forces did not lead to any significant defocusing becauseof the short propagation distance of 2 m in background plasma, butthey are expected to lead to a significant defocusing if the length ofthe neutralization section is increased by a sufficient amount.

Because the instability is resonant it is strongly affected and canbe effectively mitigated and controlled by the longitudinal focus-ing of the ion beam. This is confirmed by simulations of an NDCX-IIrelevant beam compressing with velocity tilt Δvb=vb ¼ 0:1. Thetwo-stream instability gain for an ion beam with a velocity tiltwhich produces ion beam focusing in background plasma aftertime Tf is given by [6–8]

G¼ 0:8ωpbTf ; ð9Þwhere ωpb≈ð4πe2nb=mbÞ1=2 is the ion beam plasma frequency. ForNDCX-II parameters, G≈4 with expðGÞ ¼ 55, and the initial density

perturbation grew due to instability large enough to spoil the linervelocity tilt only at the beam center, which prevented the beamfrom perfect longitudinal compression (Fig. 5). In contrast, theinstability gain during the same time Tf for an NDCX-II beam withno velocity tilt is G¼ γTf≈54 with expðGÞ ¼ expð54Þ and theinstability develops to the nonlinear saturation level well beforetime Tf and leads to longitudinal beam bunching (Fig. 1). Therefore,to avoid the effects of two-stream instability on the longitudinalbeam compression, the beam density nb or the beam compressiontime Tf have to be kept small enough to keep the instability gainfor compressing beam small, G¼ 0:8ωpbTf≲1.

References

[1] E.A. Startsev, I.D. Kaganovich, R.C. Davidson, European Physical Journal: Webof Conferences, 43, 2013, in press.

[2] A. Friedman, et al., Physics of Plasmas 17 (2010) 056704.[3] R.C. Davidson, M. Dorf, I. Kaganovich, H. Qin, E.A. Startsev, S.M. Lund, Nuclear

Instruments and Methods in Physics Research A 606 (2009) 11.[4] I.D. Kaganovich, G. Shvets, E.A. Startsev, R.C. Davidson, Physics of Plasmas 8

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Effects of beam-plasma instabilities on neutralized propagation �of intense ion beams in background plasmaIntroduction and theoretical modelNumerical simulations of NDCX-II experimentConclusionsReferences